Objectives: The living world is embedded in the physical world. Small organisms live in a world of diffusion. The rest of us require a system of ducts for transport of all the materials we need to live. Systems of branched ducts are found in lung, kidney, mammary gland, and many other organs. This project's primary objective is to understand the mechanisms of tissue dynamics that create branched systems. It also aims to clarify the best ways to work with continuum mechanical models of morphogenesis to realistically describe mechanics and transport in developing tissues. Additionally, the project will develop new numerical methods for mixture models with interfaces. We will then test these models in a real time living branching system, the early embryonic lung. Aim 1: Develop accurate, stable, efficient methods for solving mixture problems with sharp interfaces. Aim 2: Develop a suite of models of the mechanical aspects of branching epithelia. Aim 3: Determine the morphogenetic effect of the mechanics internal to a branching epithelium. Aim 4: Determine the morphogenetic effect of the mechanics external to a branching epithelium. Part of this project will be a determination of the appropriate constitutive laws, as well as quantification of the physical parameters, of the tissues involved in branching epithelia. This can only be effectively done by theory-guided experiments in conjunction with experiment-guided theory. The synergy of mechanistic mathematical modeling, sophisticated simulation, and experiment designed and analyzed in a quantitative mechanical framework will - help support or refute hypothesized mechanisms - clarify the relative roles of various physical phenomena in creating branched systems of ducts - help drive a part of biomedicine towards the future as a more exactly quantitative discipline.